In this paper, we propose a methodology for closed-loop, humanlike, task-specific reaching and grasping with redundant robot arms and low-complexity robot hands. Human demonstrations are utilized in a learn by demonstration fashion, in order to map human to humanlike robot motion. Principal Components Analysis (PCA) is used to transform the humanlike robot motion in a low-dimensional manifold, where appropriate Navigation Function (NF) models are trained. A series of grasp quality measures, as well as task compatibility indexes are employed to guarantee robustness of the computed grasps and task specificity of goal robot configurations. The final scheme provides anthropomorphic robot motion, task-specific robot arm configurations and hand grasping postures, optimized fingertips placement on the object surface (that results to robust grasps) and guaranteed convergence to the desired goals. The position and geometry of the objects are considered a-priori known. The efficiency of the proposed methods is assessed with simulations and experiments that involve different robot arm hand systems. The proposed scheme can be useful for various Human Robot Interaction (HRI) applications.